Exponential Synchronization for Reaction-diffusion Neural Networks with Mixed Time-varying Delays via Periodically Intermittent Control *

This paper deals with the exponential synchronization problem for reaction-diffusion neural networks with mixed time-varying delays and stochastic disturbance. By using stochastic analysis approaches and constructing a novel Lyapunov–Krasovskii functional, a periodically intermittent controller is first proposed to guarantee the exponential synchronization of reaction-diffusion neural networks with mixed time-varying delays and stochastic disturbance in terms of p-norm. The obtained synchronization results are easy to check and improve upon the existing ones. Particularly, the traditional assumptions on control width and time-varying delays are removed in this paper. This paper also presents two illustrative examples and uses simulated results of these examples to show the feasibility and effectiveness of the proposed scheme.


Introduction
In the past decade, there has been a great interest in various types of neural networks (for example, Hopfield neural networks, cellular neural networks, Cohen-Grossberg neural networks, bidirectional associative memory neural networks, competitive neural networks, etc.) due to their wide range of applications, such as signal processing, pattern recognition, image processing, associative memory, fault diagnosis, aerospace, defense, methods used in [21] were totally different from the corresponding previous works and the obtained conditions were less conservative.Particularly, the traditional assumptions on control width and time delay were removed.
Actually, the synaptic transmission in real neural networks can be viewed as a noisy process introduced by random fluctuations from the release of neurotransmitters and other probabilistic causes [22][23][24][25].Hence, noise is unavoidable and should be taken into consideration in modeling.On the other hand, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields.So we must consider that the activations vary in space as well as in time.From the above analysis, the stochastic noise perturbation and diffusion effects on dynamic behaviors of neural networks cannot be neglected, so the theoretical results on dynamic behaviors including stochastic disturbance and diffusion parameters are more reasonable.With respect to reaction-diffusion neural networks with stochastic perturbation, a few results about the dynamic analysis have been reported in the literature [26][27][28][29][30][31][32][33][34][35].
To the best of our knowledge, there are few results, or even no results concerning the synchronization issues for neural networks with mixed time-varying delays, stochastic noise perturbation and reaction-diffusion in terms of p-norm by using periodically intermittent control.The issues of integrating mixed time-varying delays, stochastic noise perturbation and reaction-diffusion effects into the study of synchronization for neural networks require more complicated analysis.Therefore, it is interesting to study this problem both in theories and applications.
Motivated by the above discussion, this paper is concerned with the exponential synchronization for reaction-diffusion neural networks with mixed time-varying delays and stochastic perturbation in terms of p-norm by using periodically intermittent control approach.Some examples with numerical simulations are provided to show the feasibility and effectiveness of the proposed method.
The main contribution of this paper can be summarized as follows: 1.It is the first time to establish the exponential synchronization criterion for reactiondiffusion neural networks with mixed time-varying delays and stochastic noise perturbation based on periodically intermittent control.
2. Unlike the existing results of synchronization for reaction-diffusion neural networks based on 2-norm (see [36][37][38][39]), some new and useful conditions are obtained in this paper to guarantee the exponential synchronization of the proposed neural networks under the periodically intermittent control in terms of p-norm.
3. The restrictions on periodically intermittent controller that the control width is greater than the time delay and the non-control width is also greater than the time delay are removed, which is more general than those periodically intermittent controllers given in [15][16][17].
4. A novel Lyapunov-Krasovskii functional is proposed and the restriction in [21] that the derivative of the time-varying delay should be smaller than one is removed.

5.
In [40], the authors pointed out that it is quite difficult to find a chaotic attractor for reaction-diffusion delayed neural networks.Obviously, this is an important and interesting open problem.In this paper, by using the classical implicit format solving the partial differential equations and the method of steps for differential difference equations, we find that if the parameters are appropriately chosen, the reaction-diffusion neural networks can exhibit chaotic attractors.
The organization of this paper is as follows: in the next section, problem statement and preliminaries are presented; in Section 3, a periodically intermittent controller is proposed to ensure exponential synchronization of reaction-diffusion neural networks with mixed time-varying delays and stochastic noise perturbation in terms of p-norm; numerical simulations will be given in Section 4 to demonstrate the effectiveness and feasibility of our theoretical results.We ends this work with a conclusion in Section 5.
Notation.Throughout this paper, R n and R n×m denote the n dimensional Euclidean space and the set of all n × m real matrices, respectively; the notation C 2,1 (R + × R n ; R + ) denotes the family of all nonnegative functions V (t, x(t)) on R + ×R n , which are continuously twice differentiable in x and once differentiable in t; (Ω, F, P) is a complete probability space, where Ω is the sample space, F is the σ-algebra of subsets of the sample space and P is the probability measure on F; E{•} stands for the mathematical expectation operator with respect to the given probability measure P; "sgn" is the sign function.

Problem statement and preliminaries
In this paper, we are concerned with a class of reaction-diffusion neural networks with mixed time-varying delays, which can be described by the following integro-differential equations: where i = 1, 2, . . ., n, n is the number of neurons in the neural networks; T with u i (t, x) corresponds to the state of the ith neural unit at time t and in space x; c i > 0 represents the decay rate of the ith neuron; a ij , b ij and d ij are, respectively, the connection strength, the time-varying delay connection strength, and the distributed time-varying delay connection strength of the jth neuron on the ith neuron; f j (•), g j (•) and h j (•) denote the activation functions; D ik 0 corresponds to the transmission diffusion operator along the ith neuron; 0 < τ ij (t) τ and 0 < τ * ij (t) τ * are the time-varying delay and the distributed time-varying delay along the axon of the jth unit from the ith unit, respectively; J i denotes the bias of the ith neuron.
The boundary condition of system (1) is and the initial value of system (1) is where τ = max{τ, τ * }, φ(s, x) = (φ 1 (s, x), . . ., φ n (s, x)) T ∈ C is bounded and continuous and C = C([−τ , 0] × Ω, R n ) be the Banach space of continuous functions, which maps [−τ , 0] × Ω into R n with the topology of uniform converge and p-norm (p is a positive integer) defined by Chaotic systems depend extremely on initial values, and even infinitesimal changes in the initial condition will lead to an asymptotic divergence of orbits.In order to observe the synchronization behavior of system (1), we introduce another delayed neural network, which is the response system of the drive system (1).However, the initial condition of the response system is defined to be different from that of the drive system.Therefore, the controlled response system of network (1) can be described by the following equations: h j v j (s, x) ds σ ij e j (t, x), e j t − τ ij (t), x dω j (t), where i = 1, 2, . . ., n, v(t, x) = (v 1 (t, x), . . ., v n (t, x)) T is an n-dimensional state vector of the neural networks; e(t, x) = (e 1 (t, x), . . ., e n (t, x) is the synchronization error signal; σ = (σ ij ) n×n is the diffusion coefficient matrix (or noise intensity matrix) and the stochastic disturbance ω(t) = [ω 1 (t), . . ., ω n (t)] T ∈ R n is a Brownian motion defined on (Ω, F, P), and This type of stochastic perturbation can be regarded as a result from the occurrence of the internal error when the simulation circuits are constructed, such as inaccurate design of the coupling strength and some other important parameters [41], therefore, it relies on the drive system (1).w(t, x) = (w 1 (t, x), . . ., w n (t, x)) T is an intermittent controller defined by where m = 0, 1, . .., k ij (i, j = 1, . . ., n) denotes the control strength, T > 0 denotes the control period and 0 < δ < T is called the control width.
The boundary condition and initial condition for response system (4) are given in the forms and where Subtracting ( 1) from ( 4) yields the error system as follows: σ ij e j (t, x), e j t − τ ij (t), x dω j (t), where In this paper, we give the following hypotheses: (H1) We assume that there exist positive constants L j , M j and N j such that the neuron activation functions f j , g j and h j satisfy the following conditions: where vj , vj ∈ R (j = 1, 2, . . ., n).
(H3) There exists a positive constant η ij (i, j = 1, 2, . . ., n) such that for any v1 , v2 , v1 , v2 ∈ R, and Before ending this section, we introduce some notations, the notion of exponential synchronization for reaction-diffusion neural networks ( 1) and ( 4) under periodically intermittent controller (5) based on p-norm, and some lemmas, which will come into play later on.

Exponential synchronization criterion
In this section, suitable T , δ and k ij are designed to realize exponential synchronization between reaction-diffusion neural networks (1) and ( 4) under the periodically intermittent controller (5) in terms of p-norm.For convenience, the following denotations are introduced.Let In the following, we will give an assumption ρ ij e εiτ , www.mii.lt/NAwhere ε i 0. It is easy to see that On the other hand, It follows from the assumption (H 4 ) that there exists a positive number θ i such that for all i = 1, 2, . . ., n.In a similar way, we have and G i (•) is decreasing.
We put the proof of Theorem in Appendix.
Remark 1. Ma et al. [37] investigated the synchronization problem for a class of stochastic reaction-diffusion neural networks with time-varying delays and Dirichlet boundary conditions in terms of 2-norm by using linear feedback control under the precondition that the derivative of the time-varying delay was smaller than one.Zhao and Deng studied the exponential synchronization of reaction-diffusion neural networks with continuously distributed delays and stochastic influence in terms of 2-norm based on adaptive control in [44].In [40], by using the Lyapunov functional method, many real parameters and inequality techniques, the global exponential synchronization for a class of delayed reaction-diffusion cellular neural networks with Dirichlet boundary conditions in terms of 2k-norm (integer k > 0) was discussed.In contrast, our results are derived considering the model with both discrete and distributed time-varying delays based on p-norm (p 2), which have more general application ranges.In this paper, this problem is concerned and some central criteria are derived by designing periodically intermittent controller.
Our results are more general and they effectually complement or improve the previously known results in the literature where only p = 2 or 2k were considered.Remark 2. As pointed out in [21], there are few results concerning the robust stability and robust synchronization schemes for complex networks, in particular stochastic complex networks and reaction-diffusion complex networks based on p-norm and ∞-norm using intermittent control.This motivates us to write this paper.It is the first time to establish the exponential synchronization criterion for neural networks with mixed time-varying delays, stochastic noise perturbation and reaction-diffusion effects in terms of p-norm.In this paper, the periodically intermittent control is generalized to study a more reasonable neural network model and the traditional restrictions in [15][16][17] that δ > τ and T − δ > τ are removed.
Remark 3. In Theorem 3, a novel Lyapunov-Krasovskii functional V (t, x) is employed to deal with the reaction-diffusion neural networks with mixed time-varying delays and stochastic perturbation.In the novel Lyapunov-Krasovskii functional (A.1), the integral term , such a newly introduced variable may lead to potentially less conservative results on the upper bound of the time derivative of time-varying delay.
Remark 4. The results in this paper show that, the exponential synchronization criteria on reaction-diffusion neural networks are dependent of time-varying delays, diffusion effects and stochastic noise fluctuations.Furthermore, we can see a very interesting fact, that is, as long as diffusion coefficients D ik in the system is large enough, then the assumption (H4) always can satisfy.This shows that under the boundary conditions ( 2) and ( 6), a large enough diffusion always may make the reaction-diffusion neural networks (1) and (4) globally exponentially synchronous under the intermittent controller (5) with condition (H5).

Numerical examples
In this section, by using the classical implicit format and the method of steps for differential difference equations, we give some examples with numerical simulations to illustrate the effectiveness of the theoretical results obtained above.
Example 1.For the sake of simplification, we consider a reaction-diffusion neural network model described by Clearly, it can be seen that the hypothesis (H1) is satisfied with ).The parameters of ( 9 The initial condition of drive system ( 9) is chosen as where (s, x) ∈ [−1.9, 0] × Ω.The reaction-diffusion neural network (9) with boundary condition (2) and initial condition (10) exhibits a chaotic behavior as shown in Fig. 1.
Remark 5.In [45], the authors studied the globally exponential synchronization for a class of reaction-diffusion neural networks with discrete variable delays and finite distributed constant delays based on periodically intermittent control under Dirichlet boundary conditions.Theorem 1 in [45] cannot be used to study this example with τ (t) > 1 for all t (fast-varying delay).However, after a simple computation, the conditions of Theorem 3 hold.The numerical simulations clearly verify the effectiveness of the developed periodically intermittent controller to the exponential synchronization of reactiondiffusion neural networks with mixed time-varying delays and stochastic perturbation based on p-norm.Remark 6.Note that the activation functions in [38,[46][47][48][49][50] are required to satisfy the condition for any vj , vj ∈ R (j = 1, 2, . . ., n).Obviously, this condition is stranger than the Lipstchizian condition in (H1).Hence, the results in [38,[46][47][48][49][50]] are unavailable for this example.

Conclusion
In this paper, a periodically intermittent controller has been proposed to ensure the exponential synchronization for a class of reaction-diffusion neural networks with mixed timevarying delays and stochastic noise perturbation under Dirichlet boundary conditions in terms of p-norm.The problem considered in this paper is more general in many aspects and incorporates as special cases various problems, which have been studied extensively in the literature.Some remarks and numerical examples have been used to demonstrate the effectiveness of the obtained results.
It should be pointed out that there are many published papers focusing on the synchronization problems of chaotic neural networks, but mixed time-varying delays, stochastic perturbation and reaction-diffusion effects have never been taken into consideration in terms of the synchronization issue based on p-norm for a variety of neural networks.To the best knowledge of the authors, this is the first paper incorporating mixed timevarying delays, stochastic perturbation and reaction-diffusion effects into the problem of exponential synchronization for chaotic neural networks under periodically intermittent control in terms of p-norm.

Q. Gan et al.
From condition (H4) in Theorem 3, we see that as long as feedback strength parameter k ii (i = 1, 2, . . ., n) is chosen small enough, then condition (H4) always holds.Therefore, we can obtain that there always exists an appropriate periodically intermittent control input strategy in response system (4) at all time such that drive-response systems (1) and ( 4) with boundary conditions ( 2) and ( 6) and initial conditions (3) and ( 7) are global exponential synchronization.However, the decomposing way used in (A.5)-(A.10)and the inequality technique used in (A.11) maybe increase the conservatism of the criteria on the upper bound of feedback strength k ii .So, this is a problem that we should study in the further.
In fact, due to the different parameters, activation functions and neural network architectures, which is unavoidable in real implementation, the master system and response system are not identical and the resulting synchronization is not exact and complex.Therefore, it is important and challenging to study the synchronization problems of nonidentical chaotic neural networks.Furthermore, our analysis is carried out under the assumption p 2 throughout this paper.Evidently, there is an interesting open problem concerning the exponential synchronization for non-identical reaction-diffusion neural networks with mixed time-varying delays and stochastic noise disturbance by using periodically intermittent control for p = 1 or based on ∞-norm.This will become our future investigative direction.
It follows from (A.2) the Dini derivation, it can be deduced that h * j e j (s, x) ds + n j=1 k ij e j (t, x) From the boundary conditions ( 2), ( 6) and Lemma 1, we can obtain [42] p Furthermore, it follows from (H1) and the fact Similarly, we have   Combining these two cases, we summarize that: (i) For (t, x) ∈ [0, δ) × Ω, it follows from (A.12) that E V (t, x) E V (0, x) .